Nonlinear instability scientific demonstrator for vehicle dynamics

ABSTRACT

A method for demonstrating a new scientific discovery made by the inventor about the nonlinear instability of vehicles, like aircrafts, automobiles and ocean vehicles. Said method includes a model and a three-gimbaled framework that permits said model to respond to inertial moments about the axes of which the moments of inertias are the smallest and the largest, wherein said model has restoring and damping capabilities along these two axes. Said method also comprises how to use a variable motor or a crank for controlling said model rotational motions about the intermediate principal axis of inertia with closed form formulas for the external driven frequencies and amplitudes to be used to excite the nonlinear instabilities of said model. Said model could be an aircraft, an automobile, a ship, or even a rectangular block.

FIELD OF THE INVENTION

The present invention is related to a demonstrator for education,demonstration and recreation purpose, more specifically an apparatus forsuch use as a demonstrator to show a nonlinear instability phenomenonwhich happens quite often to vehicles, such as aircrafts, automobiles,ocean vehicles, or the like. This phenomenon is related to the nonlinearinstability discovered in the book, “Nonlinear Instability and InertialCoupling Effects—The Root Causes Leading to Aircraft Crashes, Landvehicle Rollovers, and Ship Capsizes” (ISBN 978-1-7326323-0-1, to bepublished in November 2018) written also by the inventor. The presentinvention is particularly useful for training pilots to avoiduncommanded roll, pitch and yaw and to prevent Pilot-Induced-Oscillation(PIO); for training automobile operators to avoid rollovers; and fortraining captains to avoid violent ship motions including capsize.

BACKGROUND OF THE INVENTION

Vehicles like cars, ships, and aircrafts all have rolling problems. Carshave rollover; ships have rolling and capsizing; and aircrafts haveDutch roll and mysterious crashes. Is it just a coincident that allthese three major types of vehicles have the same rolling problem? Or itis not. The fact is that all these vehicles follow a same scientific lawand show the same symptom. These vehicles have existed for more than acentury. It is believed that the rolling problems have, so far,accounted for accidents and deaths in the level of millions and the costof injuries in medical care, disability and property damage in trillionsof dollars worldwide. This invention is one of several inventions theinventor has invented to deal with those dangerous rolling phenomena inorder to save lives on the roads, in the ocean, and in the sky aroundthe world by applications of a new scientific discovery about thenonlinear instability of the vehicle dynamics.

From the viewpoint of physics, these vehicles are nothing but rigid bodysystems of six degrees of freedom (three translational motions in threeperpendicular axes, i.e. forward/backward, left/right, up/down; andthree rotational motions about three perpendicular axes, often termedroll, pitch, and yaw). Believe it or not, the three rotational (roll,pitch, and yaw) motions under external moments have never been solvedanalytically without linearization approximation and as a result theyhave never been understood satisfactorily due to the fact that thegoverning equations for these motions are nonlinear which was extremelydifficult to deal with analytically. Although numerical simulations forthese motions have been obtained, the results were often difficult to beexplained because of the lack of the correct understanding of themechanism. There have been so many cases of SUV rollovers, airplanecrashes, and ship capsizing, which were hard to be explained and haveremained mysterious.

There has been a fundamental mistake made in dealing with the vehicledynamics. For a vehicle, no matter it is a car, a ship or an aircraft,the governing equations for its rotational motions (roll, pitch, andyaw) are given by Math.1 in the vector form. They were obtained based onNewton's second law of motions in a body-fixed reference frame, seereferences, SNAME: Nomenclature for treating the motion of a submergedbody through a fluid, Technical and Research Bulletin No. 1-5 (1950).d{right arrow over (H)}/dt=−{right arrow over (ω)}×{right arrow over(H)}+{right arrow over (M)},  Math. 1wherein {right arrow over (ω)}=(p,q,r)=({dot over (φ)},{dot over(θ)},ψ): the angular velocities of the vehicle; φ,θ,ψ: the roll, pitch,and yaw angle about the principal axes of inertias X, Y, Z,respectively; {right arrow over (H)}=(I_(x)p,I_(y)q,I_(z)r): the angularmomentum of the vehicle; I_(x),I_(y),I_(z): the moment of inertias aboutthe principal axes of inertias X, Y, Z, respectively (These parametersare constants in this frame); {right arrow over(M)}=(M_(x),M_(y),M_(z)): the external moments acting on the vehiclesabout the principal axes of inertia. In both the academies andindustries related to automobiles, aircrafts, and ships, the currentpractice to deal with Math. 1 is to make a linearization approximationfirst and then solve the equations because the nonlinear term −{rightarrow over (ω)}×{right arrow over (H)} is too difficult to deal with.The linearization approximation makes the nonlinear term −{right arrowover (ω)}×{right arrow over (H)} disappear, the equations then becomed{right arrow over (H)}/dt=  Math. 2

However, the equations are still considered in the body-fixed referenceframe which is a non-inertial frame. The reason for this is that theexternal moments (M_(x),M_(y),M_(z)) acting on vehicles and the momentsof inertia I_(x),I_(y),I_(z) are needed to be considered in thebody-fixed reference frame.

The fundamental mistake is that the nonlinear term −{right arrow over(ω)}×{right arrow over (H)} cannot be neglected because they are theinertial moments tied to the non-inertial reference frame which is thebody-fixed reference frame in this case. This mistake is similarly likewe neglect the Coriolis force which equals −2{right arrow over(Ω)}×{right arrow over (V)}, where {right arrow over (Ω)} is the angularvelocity vector of the earth and {right arrow over (V)} is the velocityvector of a moving body on earth. Then we try to explain the swirlingwater draining phenomenon in a bathtub. In this case, we are consideringthe water moving in the body-fixed and non-inertial reference framewhich is the earth. The Coriolis force is an inertial force generated bythe rotating earth on the moving objects which are the water particlesin this case. Without the Coriolis force, we cannot explain the motionsof the swirling water. Similarly in the vehicle dynamics, the vehicle isrotating, and we consider the rotational motions of the vehicle in thebody-fixed and non-inertial reference frame which is the vehicle itself.The difference between the two cases is that in the former the object(water particle) has translational motions ({right arrow over (V)})while in the latter the object (vehicle itself) has rotational motions({right arrow over (ω)}) but they both have the important inertialeffects which cannot be neglected because both the objects areconsidered in the non-inertial reference frames. In the former theinertial effect is the Coriolis force −2{right arrow over (Ω)}×{rightarrow over (V)} while in the latter the inertial effect is the inertialmoment −{right arrow over (ω)}×{right arrow over (H)} which are notforces but moments since we are dealing with rational motions instead oftranslational one. Without the inertial moment, we cannot explain manyphenomena which happened to aircrafts, automobiles, and ships, such asuncommanded motions of roll, pitch, and yaw for aircrafts;Pilot-Induced-Oscillation (PIO) for aircrafts; automobile rollovers; andship capsizes.

In the inventor's book, the equations Math.1 have been solvedanalytically without the linearization approximation and it was foundthat the pitch motion, without loss of generality assuming the pitchmoment of inertia to be the intermediate between the roll and yawinertias, is conditionally stable and becomes unstable in certaincircumstances. A brief summary of the findings is given below. Thegoverning equations of rotational motions of an aircraft or anautomobile under a periodic external pitch moment can be written inscalar form asI _(x) {umlaut over (φ)}+b ₁ {dot over (φ)}+k ₁φ=(I _(y) −I _(z)){dotover (θ)}{dot over (ψ)},  Math. 3I _(y) {umlaut over (θ)}+b ₂ {dot over (θ)}+k ₂θ=(I _(z) −I _(x)){dotover (φ)}{dot over (ψ)}+M ₂₁ cos(ω₂₁ t+α ₂₁),  Math. 4I _(z) {umlaut over (ψ)}+b ₃ {dot over (ψ)}+k ₃ψ=(I _(x) −I _(y)){dotover (φ)}{dot over (θ)},  Math. 5wherein b₁,b₂,b₃ are the damping coefficients for roll, pitch, and yaw,respectively; k₁,k₂,k₃ are the restoring coefficients for roll, pitch,and yaw, respectively; M₂₁ is the external pitch moment amplitude; ω₂₁and α₂₁ are the frequency and phase of the external pitch moment,respectively. These equations represent a dynamic system governing therotational dynamics of vehicles, such as an aircraft when taking off orapproaching to landing or an automobile when running off the curb wherethe most fatal rollovers happen. According to the current practice inthe industries under the linearization approximation, these equationsbecomeI _(x) {umlaut over (φ)}+b ₁ {dot over (φ)}+k ₁φ=0,  Math. 6I _(y) {umlaut over (θ)}+b ₂ {dot over (θ)}+k ₂ θ=M ₂₁ cos(ω₂₁ t+α₂₁),  Math. 7I _(z) {umlaut over (ψ)}+b ₃ {dot over (ψ)}+k ₃ψ=0.  Math. 8Therefore the current practice says that the vehicle will only havepitch motion, no roll and yaw motions because there are no momentsacting on roll and yaw directions. In reality, however, there existmoments acting in roll and yaw directions as indicated by the nonlinearterms in the right hand sides of Math. 3 and Math. 5, respectively.These moments are the components of the inertial moment vector −{rightarrow over (ω)}×{right arrow over (H)} along roll and yaw directions,respectively, and they are real and must not be neglected. Thelinearization theory assumes that these nonlinear terms are small sothat they can be neglected. The fact is that this assumption is notalways valid. The reason is explained below. The roll and yaw dynamicsystems of vehicles are harmonic oscillation systems as shown in Math. 3and Math. 5. As we know for a harmonic system, a resonance phenomenoncan be excited by a driving mechanism no matter how small it is as longas its frequency matches the natural frequency of the system. It wasfound in the inventor's book mentioned above that under certaincircumstances the nonlinear terms, (I_(y)−I_(z)){dot over (θ)}{dot over(ψ)} and (I_(x)−I_(y)){dot over (φ)}{dot over (θ)} can simultaneouslyexcite roll and yaw resonances, respectively. In these cases, the pitchmotion becomes unstable and the roll and yaw motions grow exponentiallyat the same time under the following two conditions, Math. 9 and Math.10. Such nonlinear instability is a phenomenon of double resonances,i.e. roll resonance in addition to yaw resonance.

$\begin{matrix}{{{A_{P} > A_{P - {TH}}} = {\frac{1}{\omega_{21}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}\mspace{14mu}{and}}}\mspace{14mu}{{\omega_{21} = {\omega_{10} + \omega_{30}}},}} & {{Math}.\mspace{14mu} 9} \\{{{A_{P} > A_{P - {TH}}} = {\frac{1}{\omega_{21}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}\mspace{14mu}{and}}}\mspace{14mu}{{\omega_{21} = {{\omega_{10} - \omega_{30}}}},}} & {{Math}.\mspace{14mu} 10}\end{matrix}$wherein A_(P) is the pitch response amplitude under the external pitchmoment M₂₁ cos(ω₂₁t+α₂₁); ω₁₀=√{square root over (k₁/I_(x))} andω₃₀=√{square root over (k₃/I_(z))} are the roll and yaw naturalfrequencies, respectively. The nonlinear dynamics says that the pitchmotion is stable until the pitch motion reaches the threshold valuesA_(P-TH) given in Math. 9 or Math.10. These threshold values show thatthe vehicle has two dangerous exciting frequencies in pitch. These twofrequencies are either the addition of the roll natural frequency ω₁₀and the yaw natural frequency ω₃₀ or the subtraction of them. At eachfrequency, the pitch amplitude threshold for pitch to become unstable isinversely proportional to the pitch exciting frequency, proportional tothe square root of the product of the roll and yaw damping coefficients,and inversely proportional to the square root of the product of thedifference between the yaw and pitch moments of inertia and thedifference between the pitch and roll moments of inertia. In summary,there are three factors having effects on the pitch threshold and theyare a) the roll and yaw damping, b) the pitch exciting frequency, and c)the distribution of moments of inertia. The most dominant one amongthese three factors is the damping effect since the damping coefficientscould go to zero in certain circumstances, for example, aircraft yawdamper malfunction which makes the yaw damping become zero, or aircraftin stall condition which makes the roll damping become zero. When eitherthe roll damping or the yaw damping is approaching to zero, the pitchthreshold is approaching to zero as well and the pitch motion, even itis small but as long as larger than the threshold value, will becomeunstable and transfer energy to excite roll and yaw resonances. That isthe root mechanism behind all these mysterious tragedies mentionedabove. In the inventor's book detailed scientific proofs based onanalytical, numerical and experimental results have been given. Manyreal case analyses, like aircraft crashes and SUV rollovers, have beengiven as well.

The nonlinear instability is always tied with the rotational directionwhere the moment of inertia is the intermediate between the other twoinertias. Depending on the mass distribution of a vehicle, it could haveroll, pitch, or yaw nonlinear instability if the roll, pitch, or yawmoment of inertia is the intermediate one, respectively.

There existed one device as shown in FIG. 1, fabricated in 1955 by theNACA machine shop, used as a demonstrator to show the so-called inertialroll coupling, see Richard E. Day, Coupling Dynamics in Aircraft: AHistory Perspective, NASA special publication 532 (1997). It was used toshow the effect of the inertial roll coupling due to steady rollingmotion (constant roll speed) of which the moment of inertia is thesmallest, not the intermediate. The principal roll axis of the model hasto be different with the rotation axis (the aerodynamic axes) of rollingotherwise there is no inertial coupling phenomenon, see also USAF Testpilot school, Chapter 9 Roll coupling, V. II Flying qualities phase(1986). As shown in FIG. 1, the model has to be asymmetric top andbottom and the bars with adjustable weights can be rotated in pitch tovary the product of inertia, I_(xy), and thus change the angle of theprincipal roll axis relative to the roll rotation axis. The nonlinearinstability phenomenon discovered by the inventor happens in unsteady(non-constant) rotational motion about the principal axis of theintermediate moment of inertia (not the smallest) of aircrafts.Therefore, this device cannot be used to demonstrate the nonlinearinstability phenomenon of aircrafts.

The above described nonlinear instability of vehicle dynamics, includingaircraft dynamics, automobile dynamics, ocean vehicles dynamics, or thelike, is a new scientific discovery made by the inventor. Thereforethere is a need for an apparatus which can perform experiments to showthe dangerous nonlinear instability in order to understand and preventsuch double resonances and to save lives.

SUMMARY OF THE INVENTION

The principal objective of the present invention is to provide anapparatus to prove the new scientific discovery by disclosing why, whenand how the nonlinear instability happens and to provide an educationaltool, as a demonstrator, to show the nonlinear instability to educatepilots, automobile operators, captains, etc. to prevent such dangerousphenomena.

In one embodiment, a method and an apparatus are presented foraircrafts. As an example, this apparatus provides a means for anaircraft model supported by a three-gimbal framework with restoring anddamping capabilities in roll and yaw directions to demonstrate thenonlinear pitch instability of aircrafts, assuming the pitch moment ofinertia of the aircraft model to be the intermediate between the rolland yaw inertias. The model is symmetric about its XOY and XOZ planesand able to rotate ±360° in roll and pitch, and almost ±90° in yaw, seeFIG. 2 and FIG. 3. This apparatus permits only external pitch moments tobe exerted on the model therefore any roll and yaw motions of the modelare due to the nonlinear inertial moments which have been neglected sofar in analyses in the academies and industries related to aircrafts.The natural roll and yaw frequencies and the roll and yaw dampingcoefficients of the model may be adjusted to match any full scaleaircraft's data so that the nonlinear dynamics of the full scaleaircraft is able to be simulated. By adjusting the roll and yaw dampingand applying an externally exciting pitch moment at certain frequencyand amplitude, the nonlinear pitch instability phenomenon can betriggered so that the roll and yaw resonances may be excited without anyhelp of external roll and yaw moments. The frequencies and amplitudes ofthe externally exciting pitch moments may be based on the theory in theinventor's book as briefly described above. The externally excitingpitch moments may be applied manually by hand acting on either the crankor directly on the inner frame at desired frequencies and amplitudes. Insuch a way, uncommanded roll and yaw motions, andpilot-induced-oscillations may be demonstrated.

In another embodiment, a method and an apparatus are presented forautomobiles. As an example, this apparatus provides a means for anautomobile model supported by a three-gimbal framework with restoringand damping capabilities in roll and yaw directions to demonstrate thenonlinear pitch instability of automobiles, assuming the pitch moment ofinertia of the automobile to be the intermediate between the roll andyaw inertias. The model is symmetric about its XOY, XOZ, and YOZ planesand able to rotate ±360° in roll and pitch, and almost ±90° in yaw seeFIG. 16. This apparatus permits only external pitch moments to beexerted on the model therefore any roll and yaw motions of the model aredue to the nonlinear inertial moments which have been neglected so farin analyses in the academies and industries related to automobiles. Thenatural roll and yaw frequencies and the roll and yaw dampingcoefficients of the model may be adjusted to match any full scaleautomobile's data so that the nonlinear dynamics of the full scaleautomobile is able to be simulated. By adjusting the roll and yawdamping and applying an externally exciting pitch moment at certainfrequency and amplitude, the nonlinear pitch instability phenomenon canbe triggered so that the roll and yaw resonances are excited without anyhelp of external roll and yaw moments. The frequencies and amplitudes ofthe externally exciting pitch moments may be based on the theory in theinventor's book as briefly described above. The externally excitingpitch moments may be applied manually by hand acting either on the crankor directly on the inner frame at desired frequencies and amplitudes.

In yet another embodiment, a method and an apparatus are presented forautomobiles. As an example, this apparatus provides a means for anautomobile model supported by a three-gimbal framework with restoringand damping capabilities in roll and pitch directions to demonstrate thenonlinear yaw instability of automobiles, assuming the yaw moment ofinertia of the automobile to be the intermediate between the roll andpitch inertias (this is to simulate certain truck loading conditionssuch that the yaw moment of inertia of the truck becomes theintermediate one). The model is symmetric about its XOY, XOZ, and YOZplanes and able to rotate ±360° in roll and yaw, and almost ±90° inpitch see FIG. 17. This apparatus permits only external yaw moments tobe exerted on the model therefore any roll and pitch motions of themodel are due to the nonlinear inertial moments which have beenneglected so far in analyses in the academies and industries related toautomobiles. The natural roll and pitch frequencies and the roll andpitch damping coefficients of the model may be adjusted to match anyfull scale automobile's data so that the nonlinear dynamics of the fullscale automobile is able to be simulated. By adjusting the roll andpitch damping and applying an externally exciting yaw moment at certainfrequency and amplitude, the nonlinear yaw instability phenomenon can betriggered so that the roll and pitch resonances are excited without anyhelp of external roll and pitch moments. The frequencies and amplitudesof the externally exciting yaw moments may be based on the theory in theinventor's book as briefly described above. The externally exciting yawmoments may be applied manually by hand acting either on the crank ordirectly on the inner frame at desired frequencies and amplitudes.

In still yet another embodiment, a method and an apparatus are presentedfor ocean vehicles. This apparatus provides a means for a ship modelsupported by a three-gimbal framework with restoring and dampingcapabilities in roll and pitch directions to demonstrate the nonlinearyaw instability of ocean vehicles, assuming the loading conditionleading to the yaw moment of inertia to be the intermediate between theroll and pitch moment of inertias. The model is symmetric about its XOYand XOZ planes and able to rotate ±360° in roll and yaw, and almost ±90°in pitch, see FIG. 18. This apparatus permits only external yaw momentsto be exerted on the model therefore any roll and pitch motions of themodel are due to the nonlinear inertial moments which have beenneglected so far in analyses in the academies and industries related toocean vehicles. The natural roll and pitch frequencies and the roll andpitch damping coefficients of the model may be adjusted to match anyfull scale ocean vehicle's data so that the nonlinear dynamics of thefull scale ocean vehicle is able to be simulated. The naturalfrequencies and damping coefficients of roll and pitch have taken intoaccount of the added mass effects for ocean vehicles therefore the modeldoes not need to be in water. By adjusting the roll and pitch dampingand applying an externally exciting yaw moment at certain frequency andamplitude, the nonlinear yaw instability phenomenon can be triggered sothat the roll and pitch resonances may be excited without any help ofexternal roll and pitch moments. The frequencies and amplitudes of theexternally exciting yaw moments may be based on the theory in theinventor's book as briefly described above. The external yaw moments maybe applied manually by hand acting either on the crank or directly onthe inner frame at desired frequencies and amplitudes. In such a way,ship capsizing phenomenon in following and quartering seas may bedemonstrated.

In further yet another embodiment, a method and an apparatus arepresented for aircrafts. As an example, this apparatus provides a meansfor an aircraft model supported by a three-gimbal framework withrestoring and damping capabilities in pitch and yaw directions todemonstrate the nonlinear roll instability of aircrafts, assuming theroll moment of inertia of the aircraft to be the intermediate betweenthe pitch and yaw inertias. This model is to simulate certain aircraftswhich have the roll moment of inertia to be the intermediate one betweenthe other two, such as B-52 bomber. The model is symmetric about its XOYand XOZ planes and able to rotate ±360° in roll and pitch, and almost±90° in yaw, see FIG. 21 and FIG. 22. This apparatus permits onlyexternal roll moments to be exerted on the model therefore any pitch andyaw motions of the model are due to the nonlinear inertial moments whichhave been neglected so far in analyses in the academies and industriesrelated to aircrafts. The natural pitch and yaw frequencies and thepitch and yaw damping coefficients of the model may be adjusted to matchany full scale aircraft's data so that the nonlinear dynamics of thefull scale aircraft is able to be simulated. By adjusting the pitch andyaw damping and applying an externally exciting roll moment at certainfrequency and amplitude, the nonlinear roll instability phenomenon canbe triggered so that the pitch and yaw resonances may be excited withoutany help of external pitch and yaw moments. The frequencies andamplitudes of the externally exciting roll moments may be based on thetheory in the inventor's book as briefly described above. The externallyexciting roll moments may be applied manually by hand acting on eitherthe crank or directly on the inner frame at the desired frequencies andamplitudes. In such a way, uncommanded pitch and yaw motions, andpilot-induced-oscillations may be demonstrated.

In still further yet another embodiment, a method and an apparatus arepresented for a general case of a rigid body having roll and yawrestoring and damping capabilities. As an example, this apparatusprovides a means for a rigid rectangular block model supported by athree-gimbal framework with restoring and damping capabilities in rolland yaw directions to demonstrate the nonlinear pitch instability ofsuch rigid body system. The pitch moment of inertia of the block is bydesign to be the intermediate between the roll and yaw inertias. Themodel is symmetric about its XOY, XOZ, and YOZ planes and able to rotate±360° in roll, pitch, and yaw, see FIG. 23. This apparatus permits onlyexternal pitch moments to be exerted on the model therefore any roll andyaw motions of the model are due to the nonlinear inertial moments. Byadjusting the roll and yaw damping and applying an externally excitingpitch moment at certain frequency and amplitude, the nonlinear pitchinstability phenomenon can be triggered so that the roll and yawresonances may be excited without any help of external roll and yawmoments. The frequencies and amplitudes of the externally exciting pitchmoments may be based on the theory in the inventor's book as brieflydescribed above. The externally exciting pitch moments may be appliedmanually by hand acting either on the crank or directly on the outerframe at desired frequencies and amplitudes.

In further yet another embodiment, for all the above apparatuses foraircrafts, automobiles and ocean vehicles, the cranks are replaced bydrive-control assemblies to controllably drive the nonlinear unstablemotions in a precision fashion, which includes desired frequencies andamplitudes. The drive-control assembly comprises a variable speed motor,a slider, a slotted link mechanism, a gear rack and a gear. A method foradjusting the assembly for the precision control is presented.

BRIEF DESCRIPTION OF THE DRAWINGS

The following descriptions of drawings of the preferred embodiments aremerely exemplary in nature and are not intended to limit the scope ofthe invention, its application, or uses in any way.

FIG. 1 is a perspective view of a prior related art of a demonstratorfor showing the roll inertial coupling.

FIG. 2 is a perspective view of the apparatus as a demonstrator relatedto aircrafts in accordance with the first preferred embodiment.

FIG. 3a is perspective view of the aircraft model and the roll and yawaxles without the supporting frames. This aircraft model has the pitchmoment of inertia as the intermediate between the roll and yaw inertias.FIG. 3b is a perspective view of the aircraft model assembly. FIG. 3c isthe perspective views of the roll and yaw axle assembly.

FIG. 4 is a side view of the aircraft model in FIG. 3 a.

FIG. 5 is a back view of the aircraft model in FIG. 3 a.

FIG. 6 is a top view of the aircraft model in FIG. 3 a.

FIG. 7 is a zooming-in perspective view of the restoring and dampingassemblies of the aircraft model in FIG. 3 a.

FIG. 8 is a zooming-in side view of the aircraft model in FIG. 4 withoutparts 109 a, 109 c, 110 a, 110 c, and 111 a.

FIG. 9a is a perspective view of the spring and damper assembly as oneunit and FIG. 9b is a perspective view of the spring and the damper tobe used separately.

FIG. 10 is a sectional view of the cutting plane “A-A” in FIG. 6.

FIG. 11 is a perspective view of the roll restoring and damping assembly105 of the aircraft model in FIG. 2.

FIG. 12a is an assembly view of the wing and the balance load in FIG. 3a.

FIG. 12b is a zooming-in view of the Dovetail groove sliding connectorin FIG. 12 a.

FIG. 13 is a perspective view of the apparatus with the motor-drivingassembly for precision control of the external exciting moments.

FIG. 14a is a perspective zooming-in view of the motor-driving assemblyof FIG. 13. FIG. 14b is a view of FIG. 14a without the gear rack 151 andthe slotted link 153. FIG. 14c is a side view of FIG. 14a with theaircraft model pitch down at 45°.

FIG. 15 is a cranking arm with the sliding pin.

FIG. 16a is a perspective view of the apparatus with a bus model as ademonstrator related to automobiles for the case in which the pitchmoment of inertia of an automobile is the intermediate between the othertwo inertias. FIG. 16b is the zooming-in view of the bus model withoutthe supporting frames and without the top adjustable weight 270 a shownin FIG. 16 a.

FIG. 17 is another way of mounting the model to the base in order tosimulate the case in which the yaw moment of inertia of an automobile(like some loading conditions of big trucks) is the intermediate betweenthe other two inertias.

FIG. 18 is a perspective view of the apparatus with a ship model as ademonstrator related to ocean vehicles.

FIG. 19 is a top view of the ship model of FIG. 18 without thesupporting frames.

FIG. 20 is a zooming-in perspective view of the ship model of FIG. 18without the supporting frame and with a shell section cut out amidships.

FIG. 21 is a perspective view of the apparatus with an aircraft modelwhich has the roll moment of inertia as the intermediate between theother two inertias.

FIG. 22 is the zooming-in view of FIG. 21 without the supporting frames.

FIG. 23 is a perspective view of an apparatus with a rigid body model ofrectangular block.

FIG. 24 is a perspective view of the individual elements of therestoring and damping assembly.

FIG. 25 is a side view of one of the roll restoring and dampingassembly.

FIG. 26 is a side view of one of the yaw restoring and damping assembly.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Referring to the drawings, more particularly to FIG. 2, preferred theapparatus comprises a flat round base 101, an upside-down U typesupporting frame 103 (the outer frame) rigidly mounted on the base 101,a crank 102 rigidly connected to another rectangular frame 104 (theinner frame) runs through a bearing (not shown in the figure) which ismounted in the outer frame 103 so that the crank can smoothly rotate theinner frame 104, an aircraft model 106 comprising two symmetric (top andbottom) parts about XOY plane is also symmetric (left and right) aboutXOZ plane as shown in FIG. 3a so that the principal inertial axes of themodel are along the roll, pitch, and yaw axes of the model. The top andbottom parts are kept at certain distance and connected with each otherat the two points located at the tips of wings 109 a, 109 b, 109 c, and109 d, and the yaw axles in order to create one rigid piece of aircraftmodel. The significance of the model being two parts connected only atthe two points and the yaw axles is that the empty space between the twoparts provides space for a long roll axle and for the model to yaw up toalmost ±90°. The center of gravity (COG) of the model is located atPoint O as shown in FIG. 3a and FIG. 10. The COG coincides with theintersection point of roll, pitch, and yaw axes. Therefore, the roll,pitch, and yaw axes coincide with the X, Y and Z axes of the coordinatesystem as shown in FIG. 3a . A long roll axle 108 supporting the model106 is in turn supported by two bearings (not shown) at the two ends of108 and the two bearings are mounted in the two opposite sides of theframe 104. Yaw restoring and damping assemblies 107 a and 107 b aremounted on the top part of the model, and yaw restoring and damping 107c and 107 d (not shown) on the bottom part of the model as shown in FIG.3a . A roll restoring and damping assembly 105 in FIG. 2 and FIG. 11 ismounted on one side of the frame 104. In this embodiment, the pitchmoment of inertia is the intermediate between the roll and yaw inertias,which represents the case of the most commercial aircrafts, for example,Boeing 737.

As shown in FIG. 3a , the aircraft model 106 has a top fuselage 112 aand a bottom fuselage 112 b, wings 109 a and 109 b are rigidly connectedto top fuselage and wings 109 c and 109 d to the bottom fuselage, spacer111 a connecting the top wing 109 a and the bottom wing 109 c, spacer111 b connecting the top wing 109 b and the bottom wing 109 d, thespacers are to keep the top and bottom fuselage with wings apart atenough distance to allow the model 106 to freely rotate up to almost±90° about the yaw axis (Z axis), adjustable weights 110 a, 110 b, 110c, and 110 d sliding along two grooves which are perpendicular to eachother and located on the wings and the weights, respectively, theseweights could be positioned almost anywhere on the wings to vary themoment of inertias and to tune the COG of the model to coincide with theorigin of the roll, pitch, and yaw axes, the weights 110 a, 110 b, 110c, and 110 d are identical, the roll axle 108 is along the center lineof the model and connected to the yaw axles 115 a and 115 b at the COGby a cross connector 113 as shown in FIG. 3c . As shown in FIG. 5 twoyaw restoring and damping assemblies 107 a and 107 b are mounted on thetop fuselage while two other yaw restoring and damping assemblies 107 cand 107 d are mounted on the bottom fuselage, the four spring anddamping assemblies 107 a, 107 b, 107 c, and 107 d are identical. Asshown in FIG. 3a , an axle collar 131 and a gear 125 located on eachvery ends of the roll axle 108 keep the axle fixed along the X directionwhen the model is moving. The standalone aircraft model 106 is shown inFIG. 3b . The roll and yaw axle assembly 132 is shown in FIG. 3 c.

A zooming-in perspective view of the restoring and damping assemblies107 a and 107 b on the top fuselage 112 a is given in FIG. 7. A sideview of the spring and damping assemblies 107 a and 107 c is given inFIG. 8. The assembly 107 a comprises spring and damper combinations 119a 1 and 119 a 2, a gear rack 117 a, a right angle bracket 118 a mountedon the top fuselage to serve as a guide for the linear motion of thegear rack 117 a, spring posts 120 a 1 and 120 a 4 are mounted on the topfuselage while spring posts 120 a 2 and 120 a 3 are mounted on the twoends of the gear rack 117 a. As shown in FIG. 5, the assembly 107 a islocated on one side of a gear 116 a and the yaw axle 115 a whileassembly 107 b is symmetrically located on the other side of the gear116 a and the yaw axle 115 a. As shown in FIG. 7, similarly, theassembly 107 b (also shown in FIG. 5) comprises spring and dampercombinations 119 b 1 and 119 b 2, a gear rack 117 b, a right anglebracket 118 b mounted on the top fuselage to serve as a guide for thelinear motion of the gear rack 117 b. Spring posts 120 b 1 and 120 b 4are mounted on the top fuselage while spring posts 120 b 2 and 120 b 3are mounted on the two ends of gear rack 117 b. Similar arrangements ofthe restoring and damping assemblies 107 c (as shown in FIG. 8) and 107d (as shown in FIG. 5) are located on the bottom fuselage 112 b as shownin FIG. 5. The gear 116 a is mounted on the yaw axle 115 a which goesthrough a ball bearing 114 a as shown in FIG. 7 and FIG. 10. The FIG. 10shows a section view along a plane which is cut at “A-A” of FIG. 6. Asshown in FIG. 10, the ball bearing 114 a sitting in a hole on thefuselage 112 a is suspended on the cross connector 113, the gear 116 ais fixed on the yaw axle 115 a by a screw, the yaw axle 115 a is fittedthrough the ball bearing 114 a so that the aircraft model has thefreedom to rotate about the yaw axle 115 a. The yaw axle 115 a, the gear116 a and the roll axle 108, however, do not have the degree of freedomin yaw. The aircraft model suspended on the roll axle 108 by the crossconnector 113 as shown in FIG. 8 and FIG. 10 has three rotationaldegrees of freedom (roll, pitch, and yaw). The roll axle 108 supportedat the ends by two bearings, one of which is ball bearing 126 as shownin FIG. 11. The two ball bearings are sitting in holes on the twoopposite sides of frame 104 and only one side is shown in FIG. 11. Themodel together with the frame 104 and the roll axle 108 has the freedomto rotate about the pitch axis and all three of them may be driven bythe crank as shown in FIG. 2. Therefore the aircraft model has unlimitedfreedom in roll and pitch, but only has a freedom to yaw up to almost±90° since the model can only yaw up to the points where the spacers 111a and 111 b in FIG. 3a touch the roll axle 108. Since the apparatus isdesigned to demonstrate the onset of the nonlinear instability whichhappens when the model is only pitching with certain frequencies andamplitudes, and then roll and yaw would grow from zero to large degreeswithout any help of external roll and yaw moments, the near ±90° freedomof yaw of the model is large enough for this purpose. Because anythree-gimbal framework, including the current invention, has theso-called gimbal-lock problem that is when one of the rotational axesrotates and coincides with another axis then one rotational freedom islost, the yaw motion at about ±90° of the model is not accurate anywayto simulate the free rotations of aircrafts. However, the nonlinearinstability onset is accurately simulated because the model does nothave gimbal-lock problem when roll and yaw are zero. To simulate a realaircraft dynamics in infinite freedoms in roll, pitch, and yaw, theinventor also invented a gimbal-lock-free flight simulator of which apatent will be applied separately.

When the model yaws, the yaw axles 115 a and 115 b (shown in FIG. 10)are not yawing. The gear racks 117 a, 117 b (in FIG. 7), 117 c (in FIG.8), and 117 d (not shown) are capable to slide linearly along the tracksguided by 118 a (in FIG. 7), 118 b (in FIG. 7), 118 c (in FIG. 8), and118 d (not shown), respectively. When the model yaws, the gear 116 a (inFIG. 7) will drive the gear rack 117 a and 117 b to slide, and in asimilar way, the gear 116 b (not shown in FIG. 7) will drive the gearrack 117 c and 117 d (not shown in FIG. 7) to slide. When these gearracks slide, the spring and damper combinations 119 a 1, 119 a 2, 119 b1, 119 b 2 (in FIG. 7), 119 c 1, 119 c 2 (in FIG. 8), 119 d 1 (notshown), and 119 d 2 (not shown) will extended or compressed from theoriginal setup. Therefore, the restoring and damping functions in yawcan be achieved. To keep the model symmetric about XOY and XOZ planes,these springs and dampers are identical within one model set. However,the set of springs and dampers can be changed (all eight at a time) tovary the yaw natural frequency and damping of the model. As an example,the smallest level of yaw damping that the apparatus can achieve is thecase without any dampers installed for yaw motion, meaning the springand the damper are separated as shown in FIG. 9b and only the springsare installed. In this case the yaw damping is due to the structuralfictions of the apparatus.

For roll motion, a similar restoring and damping assembly 105 (shown inFIG. 2 and FIG. 11) is installed at least on one side of the frame 104.These springs and dampers can be changed accordingly to vary the rollnatural frequency and damping of the model. The assembly 105 comprises agear rack 123, a track guide 124, a gear 125, two spring and dampercombinations 122-1 and 122-2, four spring posts 121-1, 121-2, 121-3 and121-4 as shown in FIG. 11. The gear 125 is mounted rigidly on the rollaxle 108 as shown in FIG. 11. When the model rolls, it drives the rollaxle 108 and the gear 125 to rotate with it. The gear 125 in turn drivesthe gear rack 123 to move linearly along the track guided by the bracket124. In such way, the restoring and damping in roll is achieved. A ballbearing sitting in a hole in the frame 104 to provide a smooth rotationfor the roll axle 108 is 126 as shown in FIG. 11 and another ballbearing (not shown) is located on the other end of 108. Again, thesmallest roll damping can be achieved by uninstalling the two rolldampers. In this case, the damping is due to the structural frictions ofthe apparatus.

The positions of the half cylinder weights can be adjusted on thesurface of the wings. For example, as shown in FIG. 12a , there are aDovetail groove 128 on the wing 109 a along the wingspan direction and aDovetail groove 129 along the centerline of the half cylinder weight 110a which is connected with the wing 109 a by a connector 127 which hasDovetail shapes on the top and bottom orthogonally as shown in FIG. 12b. The connector 127 is capable to slide along the groove 128 while theweight 110 a is capable to slide on the connector 127 along the groove129 and to be locked in position by a screw 130 on the wing 109 a. Anarrangement of the adjustable weight 110 b on wing 109 b is symmetric tothat on wing 109 a about the XOZ plane as shown in FIG. 3a and FIG. 6.Arrangements of the adjustable weights 110 c on wing 109 c and 110 d onwing 109 d are symmetric to that on wings 109 a and 109 b about XOYplane, respectively. The four adjustable weights 110 a, 110 b, 110 c,and 110 d are so positioned that the model is symmetric about XOY andXOZ planes and the center of gravity (COG) of the model is tuned tocoincide with the intersection point O of roll, pitch, and yaw axes asshown in FIG. 3 a.

A method for demonstrating the nonlinear instability is summarizedbelow. The crank 102 (in FIG. 2) drives the model in a harmonicoscillation in pitch which represents a control from a pilot.Oscillation pitch motion is the necessary motion happens during takeoffand landing for aircrafts. The governing equations of roll(φ), pitch(θ),and yaw(ψ), of the model 106 as shown in FIG. 3b are given as,I _(x) {umlaut over (φ)}+b ₁ {dot over (φ)}+k ₁φ=(I _(y) −I _(z)){dotover (θ)}{dot over (ψ)}+M _(x),  Math. 11θ=A ₂₁ cos(ω₂₁ t+α ₂₁),  Math. 12I _(z) {umlaut over (ψ)}+b ₃ {dot over (ψ)}+k ₃ψ=(I _(x) −I _(y)){dotover (φ)}{dot over (θ)},  Math. 13wherein, I_(x), I_(y), and I_(z) are the moment of inertias of the model106 about X, Y and Z axes, respectively, b₁ and b₃ are the dampingcoefficients for roll and yaw, respectively, k₁ and k₃ are the restoringcoefficients for roll and yaw, respectively, M_(x) is the roll momentacting on the model 106 by the roll and yaw axle assembly 132 (FIG. 3c); A₂₁, ω₂₁, and α₂₁ are the amplitude, frequency, and phase of thepitch motion driven by the crank 102, respectively.

Since the mass distribution of the assembly 132 is close to the rollaxle as shown in FIG. 3c , the roll moment of inertia of the assembly132 is very small compared with that of the aircraft model 106.Therefore the effects of the moment M_(x) is very small and may beneglected. So we assume M_(x)=0. This assumption is 100% accurate whenthe model only has pitch motions. Therefore when the crank drives themodel to rotate in pitch, the governing equations of the model 106becomes Math. 14, Math. 12, and Math.13.I _(x) {umlaut over (φ)}+b ₁ {dot over (φ)}+k ₁φ=(I _(y) −I _(z)){dotover (θ)}{dot over (ψ)}  Math. 14As we know the equations Math. 14, Math. 12, and Math.13 have thefollowing solutions,φ=0,  Math. 15θ=A ₂₁ cos(ω₂₁ t+α ₂₁),  Math. 12ψ=0.  Math. 16However, according to the theory in the inventor's book, it was foundthat the motions represented by Math. 15, Math. 12, and Math. 16 willbecome unstable, then roll and yaw motions will grow exponentially underthe conditions Math. 9 and Math. 10 described above.

To do the demonstration, the moments of inertias of the model 106 asshown in FIG. 3b may be measured before it is installed on the assembly132. After the model 106 is installed as shown in FIG. 2, the roll andyaw natural frequencies are measured by free-rotating tests to be ω₁₀and ω₃₀, respectively. For example, in a prototype of the apparatus thefrequencies were tuned to be about ω₁₀=2π=6.28 and ω₃₀=π=3.14. Thismeans that the natural roll frequency of the prototype is 1 (1/s) andthe natural yaw frequency of it is 0.5 (1/s). The damping coefficientsof roll and yaw can be also measured by free-decay tests, respectively.The preferred condition for the damping is the case in which the dampersin roll and yaw directions are not installed. This is the case of theminimum damping situations and therefore two dangerous pitch amplitudethresholds given by Math. 9 and Math. 10 are minimized as well.

When the crank drives the model in a very low frequency, ω₂₁<<|ω₁₀−ω₃₀)(a very long period), the aircraft model only shows pitch motionscontrolled by the crank, no roll and yaw motion are observed. Forexample, for the prototype this case means ω₂₁<<3.14 and the drivingperiod needs to be much longer than 2 seconds, say 4 seconds or more.When the driving frequency ω₂₁ is increased and approaching to |ω₁₀−ω₃₀|together with a pitch amplitude exceeding the threshold given by Math.10, roll and yaw double resonances should happen. In this case, violentroll and yaw motion are observed. For example in the prototype case, thefirst dangerous frequency is ω₂₁=|ω₁₀−ω₃₀|=3.14 and the pitch amplitudethreshold is found by testing to be about 90°. When maintaining the samefrequency at ω₂₁=3.14, but decreasing the pitch amplitudes to be about40° which is less than the threshold of 90°, a disappearance of the rolland yaw resonances should be observed, and the only motion remains isthe pitch motion, meaning that the pitch motion is stable. When thecrank driving frequency continues increase to the second dangerousfrequency ω₂₁=ω₁₀+ω₃₀ together with pitch amplitude exceeding thethreshold value given by Math. 9, roll and yaw resonances should happenagain. Since |ω₁₀−ω₃₀| is smaller than ω₁₀+ω₃₀, the amplitude thresholdbased on ω₂₁=|ω₁₀−ω₃₀| by Math. 10 is larger than that based onω₂₁=ω₁₀+ω₃₀ by Math. 9. [For example for the prototype case,|ω₁₀−ω₃₀|=3.14 and ω₁₀+ω₃₀=3π=9.42, so the pitch amplitude thresholdbased on ω₁₀+ω₃₀=9.42 was 3 times smaller than the pitch amplitudethreshold (about 90°) based on ω₂₁=|ω₁₀−ω₃₀|=3.14. Therefore the pitchamplitude threshold for the second frequency ω₂₁=9.42 was about 30°.When the externally excited pitch amplitude was exceeding about 30° andthe externally exciting frequency was at ω₂₁=9.42, the roll and yawdouble resonances were observed again]. When maintaining the pitchamplitude given by Math. 9 but increasing the pitch frequency to largerthan ω₁₀+ω₃₀, a disappearance of roll and yaw resonances should beobserved again, and the only motion remains is pitch in this case. Forthe prototype case, the roll and yaw resonances were excited by theinertial moments on the right hand sides of Math. 14 and Math.13,respectively. These inertial moments are nonlinear terms and neglectedin the current practice by the linearization approximation in theindustry. These phenomena observed above are consistent with thepredictions of the theory found and described in detail in the bookwritten also by the inventor.

In another embodiment as shown in FIG. 13, FIG. 14a , FIG. 14b , andFIG. 14c , the crank 102 in FIG. 2 is replaced by a more precise drivingsystem which comprises a variable speed motor 156, a crank arm 157, aslider pin 158, a slotted link 153, a gear 150, a gear rack 151, trackguides 152 a, 152 b, and 152 c, a stand 155 to support the motor 156,and a stand 154 to support the gear rack 151. The slotted link 153 isrigidly connected to the gear rack 151. The effective length of thecrank link 153 is adjustable by sliding and locking the slider pin 158along the Dovetail groove in 157 as shown in FIG. 15. The setup in FIG.13, FIG. 14a , FIG. 14b , and FIG. 14c is for demonstration purpose onlyto show how the motor drives the pitch rotation of the aircraft model.For a real experiment, the motor RPM and the position of the slider pinneed to be adjusted according to the goal of the experiment. Theseadjustments are to be done according to the following formulas, Math. 17and Math. 18. The two dangerous frequencies in terms of the RPM of themotor are calculated asω_(motor)=30|ω₁₀−ω₃₀|/π(RPM)  Math. 17ω_(motor)=30(ω₁₀+ω₃₀)/π(RMP).  Math. 18The pitch motion of the aircraft model is given asθ=(R ₁ /R ₂)sin(ω_(motor) πt/30)(rad),  Math. 19wherein R₁ is the effective length of the crank link 157 and R₂ is theradius of the gear 150. With the precise frequency and amplitude controlof the pitch motions by the motor driving system, the abovedemonstrations can be repeated in a more precision fashion.

In yet another embodiment, the apparatus can be used to demonstrate thenonlinear instability of automobiles. In this case, the aircraft modeldiscussed above is replaced by a bus model to represent an automobile asshown in FIG. 16a . The apparatus comprises a flat round base 201, anupside-down U type supporting frame 203 (the outer frame) rigidlymounted on base 201, a crank 202 rigidly connected to anotherrectangular frame 204 (the inner frame) runs through a bearing (notshown in the figure) which is sitting in a hole in the outer frame 203so that the crank can smoothly rotate the inner frame 204, a bus modelcomprising two symmetric (top and bottom) part assemblies about XOYplane as shown in FIG. 16a is also both symmetric (left and right) aboutXOZ plane and symmetric (front and rear) about YOZ plane so that theprincipal inertial axes of the model are along the roll, pitch, and yawaxes of the model. The top and bottom parts of the bus model are onlyconnected at the yaw axle and at the two points located at the middle ofthe bus by spacers 272 a as shown in FIGS. 16b and 272b (not shown)which are very similar in size like 111 a and 111 b for the aircraftmodel above. The center of gravity (COG) of the bus model is located atPoint O as shown in FIG. 16b which coincides with the intersection pointof roll, pitch, and yaw axes. Therefore the roll, pitch, and yaw axescoincide with the X, Y and Z axes of the coordinate system as shown inFIG. 16a . A long roll axle 208 (shown in FIG. 16a and FIG. 16b )supporting the bus model is in turn supported by two bearings (notshown) at the two ends of it. The two bearings are sitting in holes inthe two opposite sides of the frame 204. Yaw restoring and dampingassemblies 207 a and 207 b are mounted on the top part as shown in FIG.16b , two symmetric yaw restoring and damping assemblies 207 c (notshown) and 207 d (not shown) are mounted on the bottom part of the busmodel. Rectangular plates 212 a and 212 b of the bus model serve as thetop and bottom bases, respectively, in the same function as the top andbottom parts of the fuselage of the aircraft model above. Adjustableweight 210 a, 210 b, 210 c, and 210 d (not shown) are movable on theplates 212 a and 212 b along the Dovetail grooves, respectively. Squarepoles 271 a-1, 271 a-2, 271 a-3, and 271 a-4 as shown in FIG. 16b areused to adjust the Z position of weight 270 a which is shown in FIG. 16a(not in FIG. 16b ). Square poles 271 b-1, 271 b-2, and 271 b-3 as shownin FIG. 16b, and 271b -4 (not shown) are used to adjust the Z positionof weight 270 b as shown in FIG. 16b . With this bus model installed inthe apparatus, similar nonlinear instability phenomena as describedabove can be observed because automobiles share the same governingequations Math. 3, Math, 4 and Math.5 with aircrafts. Therefore theyhave the same symptom.

In general for most sedans, the pitch moments of inertias are theintermediate between the other two inertias. However, for some loadingconditions of some automobiles, for example big trucks, the intermediatemoment of inertia may be in the yaw direction, not in pitch. In thiscase the automobile will show nonlinear yaw instability instead ofpitch. In order to show this yaw instability, the bus model is installedin a different way as shown in FIG. 17 wherein the crank is driving themodel in yaw direction instead of pitch.

In still yet another embodiment, the apparatus can be used todemonstrate nonlinear yaw instability of ocean going vessels, forexample, containership. A ship model 306 with the yaw moment of inertiabeing the intermediate is installed on the apparatus as shown in FIG.18. In general, a ship pitch moment of inertial is the intermediate one,but in some loading conditions, a ship can be loaded so that the yawmoment of inertial becomes the intermediate, this is especially true forcontainerships. This apparatus comprises a base 301, a crank 302 goingthrough an outer frame 303 to be able to rotate an inner frame 304, asimilar roll restoring and damping assembly 305 is mounted on the innerframe 304. A top view of the ship model is given in FIG. 19. As shown inFIG. 19, the ship model comprises two parts 312 a (left) and 312 b(right). The two-parts arrangement is similar as all models discussedabove. The two parts are connected only at the pitch axle and at the twopoints along the yaw axle by the spacers 382 a and 382 b (not shown).The two parts 312 a and 312 b each have big cut-out amidships to make aroom for the pitch restoring and damping assemblies as shown in FIG. 20.The ship model again is symmetric about XOY and XOZ planes. A long rollaxle 308 connected with two pitch axles by a cross connector 313 shownin FIG. 19, similar as that shown in FIG. 3c , supports the model atcenter of the gravity. Adjustable weights 380 a-1, 380 a-2, 380 b-1, and380 b-2 are movable along Dovetail grooves 381 a-1, 381 a-2, 381 b-1,and 381 b-2, respectively on the horizontal plane while adjustableweights 310 b-1 and 310 b-2 are movable in a similar way in a verticalplane as shown in FIG. 20.

The roll and pitch restoring, and damping coefficients of the ship modelare adjustable to match a real ship data measured by free-decay tests sothat these coefficients include the added mass effects. Therefore theroll and pitch natural frequencies of the ship model are the same asthat of the ship it is simulating, respectively. To demonstrate thenonlinear yaw instability in following and quartering seas of which theroll and pitch damping coefficients are the minimum, a preferred setupis the case without any damper installed, which represents the minimumdamping effect the apparatus can achieve. Since the yaw moment ofinertial is the intermediate one, the yaw amplitude thresholds and thecritical frequencies for nonlinear instability are given as

$\begin{matrix}{{A_{Y - {TH}} = {\frac{1}{\omega_{31}}\sqrt{\frac{b_{1}b_{2}}{\left( {I_{y} - I_{z}} \right)\left( {I_{z} - I_{x}} \right)}}\mspace{14mu}{and}}}\mspace{14mu}{{\omega_{31} = {\omega_{10} + \omega_{20}}},}} & {{Math}.\mspace{14mu} 20} \\{{A_{Y - {TH}} = {\frac{1}{\omega_{31}}\sqrt{\frac{b_{1}b_{2}}{\left( {I_{y} - I_{z}} \right)\left( {I_{z} - I_{x}} \right)}}\mspace{14mu}{and}}}\mspace{14mu}{{\omega_{31} = {{\omega_{10} - \omega_{20}}}},}} & {{Math}.\mspace{14mu} 21}\end{matrix}$wherein ω₁₀ and ω₂₀ are the roll and pitch natural frequencies of a shipincluding added mass effects, respectively; b₁ and b₂ are the dampingcoefficients in roll and pitch of a ship including added mass effects,respectively; I_(x),I_(y),I_(z) are the moment of inertias in roll,pitch, and yaw of a ship in the air, respectively. The nonlinearinstability theory in the inventor's book says that the yaw motion isstable until the yaw amplitude reaches the threshold values A_(Y−TH) andat the critical frequencies ω₃₁ given in Math. 20 or Math. 21. Thereforea similar demonstration like that for the aircraft model discussed abovecan be performed using this apparatus.

In further yet another embodiment, the apparatus can be used todemonstrate the nonlinear roll instability for some type of aircrafts ofwhich the roll moment of inertia is the intermediate between the pitchand yaw inertias, for example, B52 aircraft with a longer wingspan thanthe length which makes the roll moment of inertia exceed the pitchmoment of inertia. In this case, the original aircraft model in FIG. 2is replaced by another aircraft model with longer wingspan and moreadjustable weights on the wings as shown in FIG. 21. The similarapparatus comprises an aircraft model 406, a base 401, an outer frame403, a crank 402, an inner frame 404, a pitch restoring and dampingassembly 405 and a long pitch axle 408. Again the model is symmetricabout XOY and XOZ planes and the center of gravity is adjusted to locateat the origin O of the coordinate system as shown in FIG. 22. The modelcomprises a top part 412 a and a bottom part 412 b. Again 412 a and 412b are connected only at the yaw axle and at the two points along theroll axis by spacers 411 a and 411 b as shown in FIG. 22. Adjustableweights 410 a, 410 e, 410 b, 410 f, 410 c, 410 g, 410 d, and 410 h aremoveable along the Dovetail grooves on wings 409 a, 409 b, 409 c, and409 d, and along the Dovetail grooves on the weights, respectively, byconnecting to similar connectors (not shown) like 127 in FIG. 12b .These adjustable weights are used to adjust the moment of inertias ofthe aircraft model. Similar yaw restoring and damping assemblies aremounted on the top and bottom parts of the model, for example,assemblies 407 a and 407 b on the top part.

Since the roll moment of inertia is the intermediate, the roll amplitudethresholds and the critical frequencies are given as

$\begin{matrix}{{A_{R - {TH}} = {\frac{1}{\omega_{11}}\sqrt{\frac{b_{2}b_{3}}{\left( {I_{z} - I_{x}} \right)\left( {I_{x} - I_{y}} \right)}}\mspace{14mu}{and}}}\mspace{14mu}{{\omega_{11} = {\omega_{20} + \omega_{30}}},}} & {{Math}.\mspace{14mu} 22} \\{{A_{R - {TH}} = {\frac{1}{\omega_{11}}\sqrt{\frac{b_{2}b_{3}}{\left( {I_{z} - I_{x}} \right)\left( {I_{x} - I_{y}} \right)}}\mspace{14mu}{and}}}\mspace{14mu}{{\omega_{11} = {{\omega_{20} - \omega_{30}}}},}} & {{Math}.\mspace{14mu} 23}\end{matrix}$wherein ω₂₀ and ω₃₀ are the pitch and yaw natural frequencies of theaircraft model, respectively; b₂ and b₃ are the damping coefficients inpitch and yaw of the model, respectively; I_(x),I_(y),I_(z) are themoment of inertias in roll, pitch, and yaw of the model, respectively.The nonlinear instability theory in the inventor's book shows that theroll motion is stable until the roll amplitude reaches the thresholdvalues A_(R−TH) and at the critical frequencies ω₁₁ given in Math. 22 orMath. 23. Therefore a similar demonstration like that for the aircraftmodel in FIG. 2 discussed above can be performed using this apparatus.

In further yet another embodiment, a general-case apparatus with athree-gimbaled framework is illustrated in FIG. 23. This apparatuscomprises a base 501; a crank 502; stanchions 503 a and 503 b; an outerring 504; an inner ring 505; a rectangular block 506; roll restoring anddamping assemblies 507 a and 507 b; yaw restoring and damping assemblies508 a and 508 b. The rectangular block has three different moments ofinertias such as the roll moment of inertia to be the smallest, thepitch moment of inertia the intermediate, and the yaw moment of inertiathe largest. The block is symmetrically mounted in the apparatus withthe principal roll, pitch, and yaw axes of the block to be aligned withthe roll, pitch, and yaw axles, respectively as shown in FIG. 23. A rollaxle is rigidly connected with the block and aligns with the roll axisas shown in FIG. 23. The roll axle is supported at the two ends by twobearings (not shown in FIG. 23) which are sitting in the holes on theinner ring 505 such that the block is smoothly rotate about the rollaxle. Another two bearings (not shown in FIG. 23) are sitting in theholes at the top and bottom of the outer ring 504 along the yaw axis.Two yaw axles are rigidly and perpendicularly connected to the innerring 505 at the top and bottom, respectively as shown in FIG. 23. Theyaw axles go through the bearings sitting in the outer ring 504 toprovide a smooth rotation of the block together with the inner ring 505about the yaw axis. Yet another two bearings are sitting in the holes onstanchion 503 a and 503 b in a similar way to provide smooth rotation ofthe outer ring 504 about the pitch axis. The crank 502 is rigidlyconnected to the outer ring 504 by a shaft (not shown) through a bearingin stanchion 503 a and able to apply external pitching moment on thering 504.

There are two roll restoring and damping assemblies 507 a and 507 b, andtwo yaw restoring and damping assemblies 508 a and 508 b. Each of theseassemblies includes a machined torsional spring 531, a torsional damper530, and a bearing 533 as shown in FIG. 24. The machined torsionalspring 531 is capable to be mounted and tied on rotational axle (roll oryaw) on one end by a screw 534 and on the other end the torsional springhas four cylindrical legs 532 a, 532 b, 532 c as shown in FIG. 24, and532 d (not shown) which are capable to slide smoothly into cylindricalholes (not shown) on the outer ring 504 or the inner ring 505 such thatthis end of the torsional spring has no freedom of motion transverselyon the ring (either 504 or 505) and the other end of the spring iscapable to rotate with either the roll axle or the yaw axle. Forexample, FIG. 25 shows the side view of the assembly 507 b. The spring531 is tied on the roll axle 509 b by screw 534 and the other end of 531is connected by the four legs (not shown in FIG. 25) on the outer ring505. The roll axle 509 b is rigidly fixed on one end along the rolldirection on the block 506 as shown in FIG. 23 and the other end of theroll axle 509 b is through a bearing (not shown) which is embedded inthe inner ring 505. The rotational damper 530 is mounted outside of theinner ring 505 and on the end of the roll axle 509 b as shown in FIG.25. The restoring and damping assembly 507 a is a mirror of the assembly507 b about the YOZ plane as shown in FIG. 23. As another example, FIG.26 shows the side view of the assembly 508 a. The yaw axle 510 a isthrough a bearing (not shown) which is embedded in the outer ring 504.One end of 510 a is fixed with the inner ring 505 as shown in FIG. 26.One end of the torsional spring 531 is tied on the yaw axle 510 aoutside of the outer ring 504. The other end of 531 is connected to theouter ring 504 by the four legs of the spring (not shown) so that thetorsional spring is capable to rotate only on one end with the yaw axle510 a. The four legs of the spring are free sliding in the four holes inthe ring 504. The torsional damper 530 is mounted at the far ends of theyaw axle 510 a in order to be easily taken off for the minimum dampingcase as discussed before. In this case, the damping of the system isonly due to the structural frictions of the apparatus. The restoring anddamping assembly 508 b is a mirror of the assembly 508 a about the XOYplane as shown in FIG. 23. A similar demonstration, as shown in the caseof the aircraft model in FIG. 2, about the nonlinear pitch instabilityof the rectangular block can be performed by this general-caseapparatus.

It should be understood that the detailed descriptions and specificexamples, while indicating the preferred embodiments, are intended forpurposes of illustration only and it should be understood that it may beembodied in a large variety of forms different from the one specificallyshown and described without departing from the scope and spirit of theinvention. For example, one modification may be as that the top part ofthe aircraft model in FIG. 3b may be asymmetric with the bottom part ofthe model, but the principal inertia axe of roll still aligns with theroll axe of the model and all the demonstrations described above couldbe achieved by such modified model. It should be also understood thatthe invention is not limited to the specific features shown, but thatthe means and construction herein disclosed comprise a preferred form ofputting the invention into effect, and the invention therefore claimedin any of its forms of modifications within the legitimate and validscope of the appended claims.

What is claimed is:
 1. A method of simulating three-degrees of freedomvehicle nonlinear rotational instability comprising: a. attaching avehicle model to a three-gimbaled framework, wherein said vehicle modelincludes one top piece and one bottom piece, wherein said top and bottompieces are identical and symmetric at least about the vertical planewhich contains their longitudinal centerlines when said top and bottompieces are in horizontal position, respectively, a two-axle-combinationshaft providing two orthogonal axles about which said vehicle modelrotates, wherein said top and bottom pieces are rigidly connected at twopoints, wherein said two points are located along a directionperpendicular to a two-axle plane which said two orthogonal axles form,wherein said two orthogonal axles coincide with the smallest and thelargest principal axes of said vehicle model, respectively; b. aplurality of restoring and damping mechanisms in the directions of thesmallest and the largest principal axes of inertia of said vehiclemodel, wherein said restoring and damping mechanisms comprise springs,dampers, gears, gear racks, posts, and guide brackets; c. a rotatableframe to provide controllable rotations about the intermediate principalaxis of inertia of said vehicle model, wherein said rotatable frame isrotatably coupled to supporting elements and a motor driving system; d.a plurality of identical loads slidable on said vehicle model foradjustments of moments of inertias for said vehicle model; e. whereinsaid rotatable frame is capable of rotating in oscillation fashions witha range of amplitudes and a range of frequencies to demonstrate thenonlinear instability of rotations about the intermediate principal ofaxis of said vehicle model, wherein said amplitudes and frequencies arecontrolled precisely by said motor driving system, wherein saidamplitudes cover a range given by$\left\{ {{\frac{0.5}{\omega_{10} + \omega_{30}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}},{\frac{2}{{\omega_{10} - \omega_{30}}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}}} \right\},$wherein ω₁₀ and ω₃₀ is the natural circular frequencies about thesmallest and the largest principal axes of inertias of said vehiclemodel, respectively, b₁ and b₃ are the damping coefficients about thesmallest and the largest principal axes of inertia of said vehiclemodel, respectively, I_(x), I_(y), I_(z) are the moments of inertias ofsaid vehicle model about the principal axes X,Y,Z, respectively, andwith I_(x)<I_(y)<I_(z), wherein said frequencies cover a range given by{0.1|ω₁₀−ω₃₀|, 2(ω₁₀+ω₃₀)}, wherein two dangerous frequencies arecontrolled by said motor driving system in terms of motor RPM given,respectively, byω_(motor)=30|ω₁₀−ω₃₀|π(RPM)ω_(motor)=30(ω₁₀+ω₃₀)/π(RPM), wherein ω_(motor) is revolutions of motorshaft per minute, wherein said oscillation fashions are given byθ=(R ₁ /R ₂)sin(ω_(motor) πt/30)(rad), wherein θ is a rotational motionof said rotatable frame, R₁ is an effective length of a crank of saidmotor driving system, R₂ is a radius of a gear of said motor drivingsystem, demonstrating the first frequency case of said two dangerousfrequencies by, fixing said motor RPM precisely at 30|ω₁₀−ω₃₀|/π thenchanging said amplitude when said amplitude exceeds a threshold given by${A_{P - {TH}} = {\frac{1}{{\omega_{10} - \omega_{30}}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}}},$said rotating in oscillation about the intermediate principal of axisbecoming unstable, demonstrating the second frequency case of said twodangerous frequencies by fixing said motor RPM precisely at30(ω₁₀+ω₃₀)/π then changing said amplitude when said amplitude exceeds athreshold given by${A_{P - {TH}} = {\frac{1}{\omega_{10} + \omega_{30}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}}},$said rotating in oscillation about the intermediate principal of axisbecoming unstable.
 2. A method of simulating three-degrees of freedomvehicle nonlinear rotational instability comprising: attaching athree-gimbaled framework to a vehicle model, wherein said vehicle modelincludes one top piece and one bottom piece, wherein said top and bottompieces are identical and symmetric at least about the vertical planewhich contains their longitudinal centerlines when said top and bottompieces are in horizontal position, respectively, a two-axle-combinationshaft providing two orthogonal axles about which said vehicle modelrotates, wherein said top and bottom pieces are rigidly connected at twopoints, wherein said two points are located along a directionperpendicular to a two-axle plane which said two orthogonal axles form,wherein said two orthogonal axles coincide with the smallest and thelargest principal axes of said vehicle model, respectively; a pluralityof restoring and damping mechanisms in the directions of the smallestand the largest principal axes of inertia of said vehicle model, whereinsaid restoring and damping mechanisms comprise springs, dampers, gears,gear racks, posts, and guide brackets; a rotatable frame to providecontrollable rotations about the intermediate principal axis of inertiaof said vehicle model, wherein said rotatable frame is rotatably coupledto supporting elements and a crank; a plurality of identical loadsslidable on said vehicle model for adjustments of moments of inertiasfor said vehicle model; wherein said rotatable frame is capable ofrotating in oscillation fashions with a range of amplitudes and a rangeof frequencies to demonstrate the nonlinear instability of rotationsabout the intermediate principal of axis of said vehicle model, whereinsaid amplitudes and frequencies are controlled by said crank, whereinsaid amplitudes cover a range given by$\left\{ {{\frac{0.5}{\omega_{10} + \omega_{30}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}},{\frac{2}{{\omega_{10} - \omega_{30}}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}}} \right\},$wherein ω₁₀ and ω₃₀ is the natural circular frequencies about thesmallest and the largest principal axes of inertias of said vehiclemodel, respectively, b₁ and b₃ are the damping coefficients about thesmallest and the largest principal axes of inertia of said vehiclemodel, respectively, I_(x),I_(y),I_(z) are the moments of inertias ofsaid vehicle model about the principal axes X,Y,Z, respectively, andwith I_(x)<I_(y)<I_(z), wherein said frequencies cover a range given by{0.1|ω₁₀−ω₃₀|,2(ω₁₀+ω₃₀)}, demonstrating the first frequency case ofsaid two dangerous frequencies by fixing said frequency at |ω₁₀−ω₃₀|then changing said amplitude when said amplitude exceeds a thresholdgiven by${A_{P - {TH}} = {\frac{1}{{\omega_{10} - \omega_{30}}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}}},$said rotating in oscillation about the intermediate principal of axisbecoming unstable, demonstrating the second frequency case of said twodangerous frequencies by fixing said frequency at (ω₁₀+ω₃₀) thenchanging said amplitude when said amplitude exceeds a threshold given by${A_{P - {TH}} = {\frac{1}{\omega_{10} + \omega_{30}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}}},$said rotating in oscillation about the intermediate principal of axisbecoming unstable.
 3. A method of simulating three-degrees of freedomrigid body nonlinear rotational instability comprising: a. a homogeneousrectangular block model attached to a three-gimbaled framework, whereina shaft aligns with either the smallest or the largest principal axis ofinertia of said block model, wherein said three-gimbaled frameworkcomprises an inner gimbal frame and an outer gimbal frame providingrotational freedoms for said block model; a plurality of restoring anddamping assemblies in the directions of the smallest and the largestprincipal axes of inertia of said block model, wherein said restoringand damping assemblies comprise machined springs with a plurality ofcylindrical legs, rotational dampers, and bearings, wherein saidcylindrical legs are slidable smoothly in a plurality of holes in saidgimbal frames; b. said outer gimbal frame is rotatably coupled tosupporting elements and a motor driving system to provide controllablerotations about the intermediate principal axis of inertia of said blockmodel; c. said outer gimbal frame is capable of rotating in oscillationfashions with a range of amplitudes and a range of frequencies todemonstrate the nonlinear instability of rotations about theintermediate principal of axis of said block model, wherein saidamplitudes and frequencies are controlled precisely by said motordriving system, wherein said amplitudes cover a range given by$\left\{ {{\frac{0.5}{\omega_{10} + \omega_{30}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}},{\frac{2}{{\omega_{10} - \omega_{30}}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}}} \right\},$wherein ω₁₀ and ω₃₀ is the natural circular frequencies about thesmallest and the largest principal axes of inertias of said block model,respectively, b₁ and b₃ are the damping coefficients about the smallestand the largest principal axes of inertia of said block model,respectively, I_(x),I_(y),I_(z) are the moments of inertias of saidblock model about the principal axes X,Y,Z, respectively, and withI_(x)<I_(y)<I_(z), wherein said frequencies cover a range given by{0.1|ω₁₀−ω₃₀|,2(ω₁₀+ω₃₀)}, wherein two dangerous frequencies arecontrolled by said motor driving system in terms of motor RPM given,respectively, byω_(motor)=30|ω₁₀−ω₃₀|/π(RPM)ω_(motor)=30(ω₁₀+ω₃₀)/π(RPM), wherein ω_(motor) is revolutions of motorshaft per minute, wherein said oscillation fashions are given byθ=(R ₁ /R ₂)sin(ω_(motor) πt/30)(rad), wherein θ is a rotational motionof said outer gimbal frame, R₁ is an effective length of a crank of saidmotor driving system, R₂ is a radius of a gear of said motor drivingsystem, demonstrating the first frequency case of said two dangerousfrequencies by fixing said motor RPM precisely at 30|ω₁₀−ω₃₀|/π thenchanging said amplitude when said amplitude exceeds a threshold given by${A_{P - {TH}} = {\frac{1}{{\omega_{10} - \omega_{30}}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}}},$said rotating in oscillation about the intermediate principal of axisbecoming unstable, demonstrating the second frequency case of said twodangerous frequencies by fixing said motor RPM precisely at30(ω₁₀+ω₃₀)/π then changing said amplitude when said amplitude exceeds athreshold given by${A_{P - {TH}} = {\frac{1}{\omega_{10} + \omega_{30}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}}},$said rotating in oscillation about the intermediate principal of axisbecoming unstable.
 4. A method of simulating three-degrees of freedomrigid body nonlinear rotational instability comprising: a homogeneousrectangular block model attached to a three-gimbaled framework, whereina shaft aligns with either the smallest or the largest principal axis ofinertia of said block model, wherein said three-gimbaled frameworkcomprises an inner gimbal frame and an outer gimbal frame providingrotational freedoms for said block model; a plurality of restoring anddamping assemblies in the directions of the smallest and the largestprincipal axes of inertia of said block model, wherein said restoringand damping assemblies comprise machined springs with a plurality ofcylindrical legs, rotational dampers, and bearings, wherein saidcylindrical legs are slidable smoothly in a plurality of holes in saidgimbal frames; said outer gimbal frame is rotatably coupled tosupporting elements and a crank to provide controllable rotations aboutthe intermediate principal axis of inertia of said block model; saidouter gimbal frame is capable of rotating in oscillation fashions with arange of amplitudes and a range of frequencies to demonstrate thenonlinear instability of rotations about the intermediate principal ofaxis of said block model, wherein said amplitudes and frequencies arecontrolled by said crank, wherein said amplitudes cover a range given by$\left\{ {{\frac{0.5}{\omega_{10} + \omega_{30}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}},{\frac{2}{{\omega_{10} - \omega_{30}}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}}} \right\},$wherein ω₁₀ and ω₃₀ is the natural circular frequencies about thesmallest and the largest principal axes of inertias of said block model,respectively, b₁ and b₃ are the damping coefficients about the smallestand the largest principal axes of inertia of said block model,respectively, I_(x),I_(y),I_(z) are the moments of inertias of saidblock model about the principal axes X,Y,Z, respectively, and withI_(x)<I_(y)<I_(z), wherein said frequencies cover a range given by{0.1|ω₁₀−ω₃₀|,2(ω₁₀+ω₃₀)}, demonstrating the first frequency case ofsaid two dangerous frequencies by fixing said frequency at |ω₁₀−ω₃₀|then changing said amplitude when said amplitude exceeds a thresholdgiven by${A_{P - {TH}} = {\frac{1}{{\omega_{10} - \omega_{30}}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}}},$said rotating in oscillation about the intermediate principal of axisbecoming unstable, demonstrating the second frequency case of said twodangerous frequencies by fixing said frequency at (ω₁₀+ω₃₀) thenchanging said amplitude when said amplitude exceeds a threshold given by${A_{P - {TH}} = {\frac{1}{\omega_{10} + \omega_{30}}\sqrt{\frac{b_{1}b_{3}}{\left( {I_{z} - I_{y}} \right)\left( {I_{y} - I_{x}} \right)}}}},$said rotating in oscillation about the intermediate principal of axisbecoming unstable.